L(s) = 1 | + 0.536·2-s − 1.71·4-s − 2.57·7-s − 1.99·8-s + 5.14·11-s + 1.47·13-s − 1.38·14-s + 2.35·16-s + 0.687·17-s − 8.09·19-s + 2.75·22-s + 0.372·23-s + 0.791·26-s + 4.40·28-s − 0.0356·29-s − 4.48·31-s + 5.24·32-s + 0.369·34-s − 1.90·37-s − 4.34·38-s + 5.16·41-s + 11.4·43-s − 8.80·44-s + 0.199·46-s + 8.52·47-s − 0.378·49-s − 2.52·52-s + ⋯ |
L(s) = 1 | + 0.379·2-s − 0.856·4-s − 0.972·7-s − 0.704·8-s + 1.55·11-s + 0.409·13-s − 0.368·14-s + 0.588·16-s + 0.166·17-s − 1.85·19-s + 0.588·22-s + 0.0777·23-s + 0.155·26-s + 0.832·28-s − 0.00662·29-s − 0.806·31-s + 0.927·32-s + 0.0632·34-s − 0.313·37-s − 0.704·38-s + 0.807·41-s + 1.74·43-s − 1.32·44-s + 0.0294·46-s + 1.24·47-s − 0.0541·49-s − 0.350·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 0.536T + 2T^{2} \) |
| 7 | \( 1 + 2.57T + 7T^{2} \) |
| 11 | \( 1 - 5.14T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 - 0.687T + 17T^{2} \) |
| 19 | \( 1 + 8.09T + 19T^{2} \) |
| 23 | \( 1 - 0.372T + 23T^{2} \) |
| 29 | \( 1 + 0.0356T + 29T^{2} \) |
| 31 | \( 1 + 4.48T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 - 5.16T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 8.52T + 47T^{2} \) |
| 53 | \( 1 + 9.12T + 53T^{2} \) |
| 59 | \( 1 - 0.176T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 + 0.0834T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 4.95T + 79T^{2} \) |
| 83 | \( 1 - 9.36T + 83T^{2} \) |
| 89 | \( 1 - 0.0123T + 89T^{2} \) |
| 97 | \( 1 - 7.62T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84443997090707383428631744011, −6.83354940521092444628789327732, −6.22366566012554908085532684074, −5.81070549216824342743700835688, −4.66814662067972552002100309359, −3.96934842194559594986582699657, −3.61247743493886317905148827006, −2.51203499443011603515690191934, −1.21513615117073993438403802641, 0,
1.21513615117073993438403802641, 2.51203499443011603515690191934, 3.61247743493886317905148827006, 3.96934842194559594986582699657, 4.66814662067972552002100309359, 5.81070549216824342743700835688, 6.22366566012554908085532684074, 6.83354940521092444628789327732, 7.84443997090707383428631744011